First Order & Second-order Systems
 

A first order system can be written in the time domain as:

The laplace transform of this equation can be given as:

K is the gain, and tau is the time constant.
 
 



 

A second order system can be written in the time domain as:

The laplace transform of this equation yields:

K is the process gain, tau is the characteristic time constant, and zeta is the dampening coeficeint.  Depending upon the value of zeta, the secon-order system forced with a step input can exhibit underdamped, critically damped, or underdamped behavior.